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You can define sets in terms of predicates (x \elem Fish \iff: fish(x)) or the other way (fish(x) \iff: x \elem Fish). So while you might intuitively say that x \elem Fish is a categorisation, it has a brother thats intuitively has the form of a fact, and you cant change one without the other.
There is not a logical distinction between intensional and extensional definitions, except
in modal logics, where it exists but depends entirely on details in your semantic setup that are in no way constrained by evidence, or
in model theory, there are intensions and extensions - but still no intensional or extensinal definitions in the underlying.
This is another manifestation of the problem.
I had (1) in mind when making up the example.
No. Its just an example of how one of the things he lists as "facts" might be redefined with just as good a justification as "fish". The problem is that when you decide what the optimal most convenient way to categorise is, you need to have some facts based on which to make that decision. But if theres no difference between facts and categorisations, then you dont.
Calling either of those two a "fact" does not seem right to me - it seems like it's conflating the signifier (the word "fish") with the signified (whatever factual basis is being used for defining the category). Now, if your objection is that we have no way of directly interacting with facts except by making categories that refer to them, whether these categories are complex high-level things like "fish" or more low-level ones like "hair follicles" or "adenine", and we have just arbitrarily elevated some of these categories to being "facts", that objection is philosophically fair - but, I would argue, not relevant in practice: a category/signifier is a better proxy for a fact/signified the more likely every existing and hypothetical human is to agree on its extension, and the great triumph of reductionism is that as categories become lower-level/further removed from day-to-day experience, they empirically become better proxies. Democrats and Republicans might have great disagreements about what is a "woman", and moderns and ancients might have disagreements about what is a "fish", but they will mostly agree about X and Y chromosomes looking different under a microscope, and would acknowledge that a hair is a hair if shown to them.
This is consistent with a model where even though we are cursed with only being able to use "categories" and not "facts", talking about reality is actually easy, and all that gets in the way is motivated reasoning, which can be dealt with by picking "categories" that are weird enough that the monkey brain fails to backpropagate its motivation to them. In such a setting, all we need to do to have access to something that is as good as facts (in the sense of behaving as if it were an aspect of objective reality, so two people with sufficient observation and discussion will always come to agree on its extension) is to pick a rich enough category of such monkey-proof signifiers, and gatekeep it. It doesn't matter if there are some grey-zone categories that are not quite factual in that sense, if we can just treat them as non-facts, and ground all our definitions in the gatekept category of facts.
Well, I'm using a mishmash of mathematical terminology, old and busted analytical philosophy and whatever schlock comes out of my badly trained neural net here, but the most serious logic I've done was in the context of computability theory, where the distinction is made and absolutely matters - people in that subculture like to build their whole notion of ontology around equality testing, and equality in intension and equality in extension are not the same. Going meta, the extension of an "intensional definition" is then all entities that have the same (or equivalent under some notion, if you are a potential mark for the HoTT pyramid scheme) syntactical definition, while the extension of an "extensional definition" is all entities whose definition has the same extension.
Okay, that clarifies it, but why would they do that? Does it help them produce good leather, perhaps because they want to batch-process all stock they get from the fishing guild together? If it actually produces better outcomes, then there shouldn't be a problem with them saying that whales are fish. If they do in fact wind up saying that whales have no hair, that seems like an instance of what I called monkey brain backpropagation of motivations further above. Why would they do that instead of just saying that as far as they are concerned, fish sometimes do have hair, but they are still going to process all fish together? Did someone pressure them to adopt the "fish have no hair" definition?
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